Continuous dependence and stability for a class of fractional partial differential equations with multiple parameters

In this paper, we studied the continuous dependence and stability of solutions for a class of fractional partial differential equations with multiple spatially varying coefficient parameters. The nonlocal operator was defined by a symmetric kernel, yielding a self-adjoint structure essential to the analysis. Using variational methods and the Minty–Browder theorem, we established the existence and uniqueness of weak solutions in the energy space $X_0$ for each admissible parameter vector $w$. We extended single-parameter stability to a multi-parameter framework by proving that the solution operator $S_f$ is continuous with respect to $w$ in the product space $\prod _i{L}^{q_i}(\mathrm{\Omega })$. Moreover, we derived an explicit global Lipschitz estimate for $S_f$ and showed its Gâteaux differentiability under mild regularity assumptions on $f(x,u,w)$. Numerical simulations confirmed continuity, Lipschitz stability, and differentiability of $S_f$ with respect to all parameters. These results provided rigorous guarantees for inverse problems and uncertainty quantification in multi-parameter fractional PDE models.